Dynamic Complexity of Planar 3-Connected Graph Isomorphism

We consider the following problem which is motivated by two different contexts independently, namely graph theory and combinatorial optimization.Given a 3-connected planar graph G with n vertices, is there a spanning closed walk W with at most 4n/3 edges?In graph theory, the above question is motivated by the famous hamiltonian result by Tutte in 1956 which says that every 4-connected planar graph is hamiltonian (a simpler proof is given by Thomassen in 1983). What happens if we relax the 4-connectivity? There is a 3-connected planar graph that is not hamiltonian, but how about a spanning close walk (which is exactly a traveling salesman tour. Sometimes such a walk is called hamiltonian walk)? How many edges are necessary to cover all the vertices of a 3-connected planar graph by a closed walk? This is exactly the above question.In combinatorial optimization, the famous traveling salesman problem in metric graphs is one of most fundamental NP-hard optimization problems. In spite of a vast amount of research several important questions remain open. In particular, the best known upper bound is not believed to be best possible. A promising direction to improve this approximation guarantee, has long been to understand the power of a linear program known as the Held-Karp relaxation [11]. On the one hand, the best lower bound on its integrality gap (for the symmetric case) is 4/3 and indeed the famous (so called 4/3-)conjecture said that this lower bound would be tight [10]. Goemans pointed out that there is a planar graph that achieves this bound. So he brought attention to the above question, i.e, the famous 4/3-conjecture is always true for 3-connected planar graphs.We prove the above problem in the following strong form;Given a circuit graph (which is obtained from a 3-connected planar graph by deleting one vertex) with n vertices, there is a spanning closed walk with at most 4(n − 1)/3 edges such that each edge is used at most twice.Moreover, our proof is constructive (and purely combinatorial) in a sense that there is an O(n2) algorithm to construct, given a 3-connected planar graph, such a walk.We shall construct an example that shows that the bound 4(n − 1)/3 is essentially tight. We also point out that 2-connected planar graphs may not have such a walk, as K2,n−2 shows.

For two integers m,n with m ≤ n, an [m, n]-factor F in a graph G is a spanning subgraph of G with m ≤ dF(v) ≤ n for all v ∈ V(F). In 1996, H. Enomoto et al. proved that every 3-connected planar graph G with dG(v) ≥ 4 for all v ∈ V(G) contains a [2,3]-factor. In this paper we extend their result to all 3-connected locally finite infinite planar graphs containing no unbounded faces.

Whitney's theorem states that every 3-connected planar graph is uniquely embeddable on the sphere. On the other hand, it has many inequivalent embeddings on another surface. We shall characterize structures of a $3$-connected $3$-regular planar graph $G$ embedded on the projective-plane, the torus and the Klein bottle, and give a one-to-one correspondence between inequivalent embeddings of $G$ on each surface and some subgraphs of the dual of $G$ embedded on the sphere. These results enable us to give explicit bounds for the number of inequivalent embeddings of $G$ on each surface, and propose effective algorithms for enumerating and counting these embeddings.Comment: 19 pages, 12 figures

The Clar number of a (hydro)carbon molecule, introduced by Clar (The aromatic sextet, 1972), is the maximum number of mutually disjoint resonant hexagons in the molecule. Calculating the Clar number can be formulated as an optimization problem on 2-connected plane graphs. Namely, it is the maximum number of mutually disjoint even faces a perfect matching can simultaneously alternate on. It was proved by Abeledo and Atkinson (Linear Algebra Appl 420(2):441–448, 2007) that the Clar number can be computed in polynomial time if the plane graph has even faces only. We prove that calculating the Clar number in general 2-connected plane graphs is \(\mathsf {NP}\)-hard. We also prove \(\mathsf {NP}\)-hardness of the maximum independent set problem for 2-connected plane graphs with odd faces only, which may be of independent interest. Finally, we give an exact algorithm that determines the Clar number of a given 2-connected plane graph. The algorithm has a polynomial running time if the length of the shortest odd join in the planar dual graph is fixed, which gives an efficient algorithm for some fullerene classes, such as carbon nanotubes.

Prescribed edges and forbidden edges for a cycle in a planar graph

Geometric routing by using virtual locations is an elegant way for solving network routing problems. Greedy routing, where a message is simply forwarded to a neighbor that is closer to the destination, is a simple form of geometric routing. Papadimitriou and Ratajczak conjectured that every 3-connected plane graph has a greedy drawing in the $\mathcal R^2$ plane [10]. Leighton and Moitra settled this conjecture positively in [9]. However, their drawings have two major drawbacks: (1) their drawings are not necessarily planar; and (2) Ω(n logn ) bits are needed to represent the coordinates of their drawings, which is too large for routing algorithms for wireless networks. Recently, He and Zhang [8] showed that every triangulated plane graph has a succinct (using O (logn ) bit coordinates) greedy drawing in $\mathcal R^2$ plane with respect to a metric function derived from Schnyder realizer. However, their method fails for 3-connected plane graphs. In this paper, we show that every 3-connected plane graph has drawing in the $\mathcal R^2$ plane, that is succinct, planar, strictly convex, and is greedy with respect to a metric function based on parameters derived from Schnyder wood.

We study the computational complexity of the isomorphism and equivalence problems on systems of equations over a fixed finite group. We show that the equivalence problem is in P if the group is Abelian, and coNP-complete if the group is non-Abelian. We prove that if the group is non-Abelian, then the problem of deciding whether two systems of equations over the group are isomorphic is coNP-hard. If the group is Abelian, then the isomorphism problem is graph isomorphism hard. Moreover, if we impose the restriction that all equations are of bounded length, then we prove that the isomorphism problem for systems of equations over finite Abelian groups is graph isomorphism complete. Finally we prove that the problem of counting the number of isomorphisms of systems of equations is no harder than deciding whether there exist any isomorphisms at all.KeywordsComputational ComplexityAbelian GroupPolynomial TimeRegular SemigroupEquivalence ProblemThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The object of this thesis is to investigate the Reconstruction Problem for planar graphs. This study naturally leads to related topics concerning certain nonplanar graphs and the use of their embeddings on appropriate surfaces to reconstruct them. The principal aim of this work is to find new techniques of reconstruction and to increase the number of classes of graphs known to be reconstructible. In achieving this aim, various important properties of graphs, such as connectivity and uniqueness of embeddings, are explored, and new results on these topics are obtained. Part I, which consists of three chapters, contains a historic a l, non-technical introduction and general graph-theoretical definitions, notation and results. Some new concepts in reconstruction are also presented, notably the idea of reconstructor sets. Part II of the thesis deals with the vertex-reconstruction of maximal planar graphs: Chapter 4 is concerned with the vertex-recognition of maximal planarity, whereas Chapter 5 deals with the vertex-reconstruction. Part III deals with edge-reconstruction: planar graphs with minimum valency 5 and 4-connected planar graphs are reconstructed in Chapters 6 and 7 respectively. In Chapter 7, extensive use is made of the concept of reconstructor sets introduced in Chapter 3. This chapter also contains a brief discussion on the reconstruction of graphs from edge-contracted subgraphs, a problem which, in certain cases, can be regarded as dual to the Edge-reconstruction Problem. Part IV is concerned with extending the results and techniques of the previous chapters to nonplanar graphs. Chapter 8 discusses where the previous techniques fa il, and indicates where new methods are needed. In Chapter 9, all graphs which triangulate some surface and have connectivity 3 are edge-reconstructed. Certain graphs which triangulate the torus or the projective plane are also shown to be weakly vertex-reconstructible. Chapter 10 deals with the edge-reconstruction of all graphs which triangulate the projective plane. The Appendix proves a conjecture of Harary on the cutvertex-reconstruction of trees. One technique used here ties up with a method employed in previous chapters on edge-reconstruction.

Improved visibility representation of plane graphs

Spanning closed walks and TSP in 3-connected planar graphs

Structure of edges in plane graphs with bounded dual edge weight

The concept of dependence of subgraphs of a plane graph is defined, as a measure of how much they overlap. It is shown that if M is a 3-connected plane graph, then the number of copies of M in a plane graph which are dependent on a given copy is bounded above by a constant c (M). The number of copies of M in any n-vertex plane graph is at most nc (m).

Given a weighting of all elements of a 2-connected plane graph G = (V,E, F ), let f(α) denote the sum of the weights of the edges and vertices incident with the face α and also the weight of α. Such an entire weighting is a proper face colouring provided that f(α) = f(β) for every two faces α and β sharing an edge. We show that for every 2-connected plane graph there is a proper face-colouring entire weighting with weights 1 through 4. For some families we improved 4 to 3.

Non-Hamiltonian Cubic Planar Graphs Having Just Two Types of Faces

On external-memory MST, SSSP and multi-way planar graph separation